The points A and B have position vectors 2i + 6j – k and 3i + 4j + k respectively. The line l passes through both A and B. Find a vector equation for the line l.

First you need to find the vector AB. This is equal to -OA+OB.

OA and OB are equal to the position vectors of A and B respectively so

 -OA+OB=  -2i - 6j + k + 3i + 4j + k = i - 2j + 2k = AB 

Then we can take any point on the curve, P, and any point on the curve can be written in the form,

P+c(i - 2j + 2k) where c is an arbitrary constant, we will take P=A so the vector equation of the line l is r= 2i + 6j – k + c(i - 2j + 2k) where c is an arbitrary constant.